The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in my interests. More specifically, what results are known of the form "the forcings which preserve $\Gamma$-determinacy are exactly the following: . . ." for $\Gamma$ some reasonably natural pointclass? I am, for the purposes of this question, taking my base set to be $\mathbb{N}$ (that is, all payoff sets are subsets of $\mathbb{N}^\mathbb{N}$).

What I've been able to figure out so far (which is not very much):

 - Continuum-closed forcings preserve all determinacy assumptions. This is just because continuum-closed forcings add no new sets of reals - hence no new payoff sets, no new points in old payoff sets, and no new strategies.

 - Countably closed forcings preserve PD (projective determinacy). A countably closed forcing adds no new reals, and hence preserves the truth of analytic formulas with parameters (=definitions of projective sets), and since no new reals are added, no new strategies are added either.

 - Countably closed forcing is *not* enough to preserve AD. The usual construction of a non-determined game can be reformulated as a countably closed forcing construction over a model of ZF; and even if the ground model satisfies AD, the generic extension will have a non-determined game.

I have tried to figure out whether either of these results reverse, but I've had no success here. The way I would attempt to phrase such a reversal would be something like the following:

(*) If $\mathbb{P}$ is some poset without property $P$, then there is a transitive model of set theory $W$ containing $\mathbb{P}$ and satisfying $\Gamma$-determinacy such that forcing with $\mathbb{P}$ over $W$ does not preserve $\Gamma$-determinacy.

So, in addition to the main question, I have the following subquestions:

(i) Are any results along the lines of (*) known?

(ii) What methods seem like they could be useful for proving results along the lines of (*)?

(iii) For that matter, is my reasoning in the bullet points above correct? It seems straightforward enough, but I've been very wrong about these sorts of things before.

Thanks in advance!

[EDIT: I forgot to mention this initially, but for the purposes of this question I'm assuming the consistency of arbitrary large cardinals, although I am very interested in how much large cardinal strength any answers require.]